It seems evident that the middle definition, "a feeling that you [and your needs] are more important or better than other people" is the attribute which the Bible warns us of.

In the New Testament the word "pride" is found only three times and each of these comes a different Greek word. The word "proud" appears more often and is usually the Greek word huperephanos, which means "showing oneself above others."

Pride then is self-centredness, selfishness, self-focus, self-worship, the "feeling that you are more important or better than other people."

When I've observed animals, even those who mate for life, they work to ensure they have their needs met ahead of their mate. They fight others off the kill, the seeds, or the dish of food until they have fed or gorged themselves only then will they allow others to feed.

God created us in His image and likeness. God has, throughout the Bible, presented Himself as giving, caring for others more than Himself even to death:

For God so loved the world, that He gave His only begotten Son (Joh 3:16 NAU)

God demonstrates His own love toward us, in that while we were yet sinners, Christ died for us. (Rom 5:8 NAU)

"A feeling that you are more important or better than other people."

What character quality did Jesus exhibit and admonish us to do so as well? Was it not love as represented by the Greek word agapao or agape as we know it?

Agape ... is not a feeling; it's a motivation for action that we are free to choose or reject. Agape
is a sacrificial love that voluntarily suffers inconvenience,
discomfort, and even death for the benefit of another without expecting
anything in return. http://www.compellingtruth.org/agape-love.html

Agape is defined using the terms like "self-sacrificing" with the principle of esteeming others ahead of yourself.

Agape is not a feeling, it is a decision, a determination to to consider the needs of others ahead of our own.

Agape is doing what God requires of us rather than doing what our 'flesh' (mind and body) demand we do.

"Truly, truly, I say to you, unless a grain of wheat falls into the earth and dies, it remains alone; but if it dies, it bears much fruit. (Joh 12:24 NAU)

Jesus here is speaking of dying to self. A fruitful life is one not ruled by self-centredness but, by dying to self issues, ruled by a desire to serve others.

whoever wishes to become great among you shall be your servant (Mat 20:26 NAU)

Those among us who aspire to be great will evidence themselves as those who joy in serving others.